Reddit mentions of A Primer of Infinitesimal Analysis

There are very few true textbooks - i.e. books designed to teach the material to those who don't already know the classical versions - written in this style.

• Bridger: Real Analysis - A Constructive Apporach
  
• Bell: A Primer of Infinitesimal Analysis - not constructive per se, but an exposition of non-classical mathematics.
  
• Vickers: Topology via Logic
  
• Lombardi: Commutative Algebra - the odd-one-out in that one does need a good working knowledge of classical commutative algebra to read the book, but it presents a whole lot of genuinely new material textbook-style.
  
• Taylor: Practical Foundations of Mathematics - Chapter 3 is an intro to constructive order theory.
  
• Troelstra: Choice Sequences - again, hardly constructive, but a good exposition of an important part of intuitionistic mathematics.
  
  While we're at it, a quick skim through the algebra chapter of Troelstra: Constructivism in Mathematics, vol. 2 should explain why there are no textbooks on abstract algebra written in the purely constructive tradition.

> Pretending that dt is a variable

Infinitesimals are not variables, and you don't have to "pretend" they're variables to do the sort of manipulation the OP mentioned. There's an excellent book on this subject by John Bell, taking the nilpotent square approach to infinitesimals. What the OP did turns out to be valid.

One way of doing "non-standard" analysis that I think closely models what a physicist does it to posit the existence of "nil-square infinitesimal numbers" which have the following property: For any infinitesimal dx, for any number a, and any smooth function f, f(a + dx) = f(a) + f'(a) * dx. Furthermore, infinitesimals dx have the property that dx^2 = 0. You can derive most rules of calculus this way. E.g.,

(f ° g)(a + dx) = f(g(a) + g'(a) * dx) = f(g(a)) + f'(g(a)) * g'(a) * dx so (f ° g)'(a) = f'(g(a)) * g'(a).

If you want to work with higher order approximations you can use infinitesimals whose cube, fourth power, etc. is 0 instead of those whose square is 0. This is the approach of synthetic differential geometry. You can read a good intro here and check out this book if you want more.

No problem. For smooth infinitesimal analysis, there's an easy-to-read introduction by John Bell. For nonstandard analysis, on the other hand, there's Keisler.

oh, sorry. i sort of misread your question. however, i would still recommend it to you given your background. i would even recommend reading certain parts of it to someone who already knows category theory. it starts off rather basic, but it does get serious and doesn't short the material at all. i have a master's in math, and it's what i started out with. i still don't know much serious category theory, but what intuition i have almost completely comes from that book. it's very good, and you're probably at the perfect level of math to get a good deal from it. in general with category theory, the more math you know, the more you'll get out or understand of category theory.

william lawvere, one of the co-authors, is one of the giants in category theory, and is the originator of smooth infinitesimal analysis and synthetic differential geometry. those are subjects you also might be interested in. a primer of infinitesimal analysis by john bell is my recommendation there. you might start of with reading his invitation to smooth infinitesimal analysis [pdf].

the category for the sciences book is also very good. it covers the material at varying levels of abstraction.

Of course! If you wanna read more, check out John Bell's Primer on Smooth Infinitessimal Analysis.

https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bell/dp/0521887186